Following [1]}, we use the \(ELPD_y\)  of the Poisson data (where \(ELPD_y\)  is defined in a similar way to \(ELPD_z\)  in (REF )) as
estimated by WAIC [2]} to select the \(\delta \) -SMI distribution \({\tilde{p}}^{(k)}_{\delta ^*}\)  with posterior predictive distribution most closely matching the true generative model, and compare against \(\eta \) -SMI, with \(\eta ^*\)  chosen in the same way. Nested MCMC targeting the the \(\delta \) -SMI posterior was implemented using STAN [3]}.
